Tower Local Control Trainee reported an aircraft failed to land and hold short of an intersecting runway while another aircraft had entered the landing threshold of the intersecting runway. The Tower Controller then instructed the aircraft on final approach to go around.

Narrative

I was training on Local. Aircraft Y was on final for Runway XX and was instructed in the air to land and hold short of Runway XY. Read back instructions and was advised of traffic on Runway XY. Aircraft X was on final for [the] runway; a tie for Runway XX Aircraft Y. He was advised Aircraft Y was cleared to land Runway XX and that traffic would be holding short of his runway at all times. Aircraft Y landed and when observed at taxi speed was told to turn left at Taxiway X; which is south of Runway XY; and aircraft read back the instruction. I go to look back at Aircraft X; which was crossing the threshold and about to touchdown when the Tower team sees that the Aircraft Y had continued through the intersection and was now on Runway XY at the Runway XX crossing intersection. Immediately told Aircraft X to go around; as he just touched his front gear down onto the runway. Aircraft X executed go-around and the Aircraft Y ended up turning left at the departure end of Runway XX; all without further incident. Staffing is incredibly low and short at ZZZ. A standalone Controller in Charge would have been helpful or more people for breaks would have also been helpful. This happened at the beginning of my training hit; and I had to continue to work for a total of 1.47 hours; more than an hour of training still after this event occurred. Aircraft X going around caused other airspace issues; and radar was super short-staffed at the time as well. If volume hadn't been so high and complex; perhaps use of LAHSO (Land and Hold Short Operations) would not had been necessary either. However; the pilot read everything back and should have told us if he was unable to comply. ZZZ always seems to be short-staffed and while training in such a high volume facility; it is going to end up being a serious safety event one day because we are pushing ourselves too hard for too long on a daily basis. ZZZ needs more staffing.